C*-algebras and their automorphism groups by Gert Kjaergard Pedersen
By Gert Kjaergard Pedersen
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The answer is: add 1st and 2nd rows to 3rd. You may think of this as two operations. I think of it as one: We can now revisit the original problems of this chapter: What makes them just a little more challenging than the problems already discussed is that they involve more than one or two steps to achieve the goal. For the first problem, the column operations are: a). divide 3rd column by , b). add 3rd column to 1st column, c). subtract 4th column from 1st, and d). multiply 4th column by . Thus the known matrix is obtained as follows (showing two steps at a time) For the second problem, the row operations are: a).
The elements of any basis of the null space can be considered the corresponding eigenvectors. This is so, because for a vector in the null space, by definition For example, for the matrix the vector is an eigenvector corresponding to . Feature 5. The eigenvalues of a diagonal matrix appear on the diagonal. That's clear, because the determinant of a diagonal matrix is the product of the diagonal entries, therefore, the characteristic polynomial of is . Armed with these insights, we can now determine the eigenvalues of the originally posed matrix a).
As you work out these exercises, note that this rule applies to all row column permutations including the swap. Exercise 62 Evaluate the matrix product Solution Exercise 63 Evaluate the matrix product Solution Exercise 64 Evaluate the matrix product Solution Exercise 65 Evaluate the matrix product Solution Exercise 66 Evaluate the matrix product Exercise 67 Evaluate the matrix product Solution Exercise 68 Evaluate the matrix product Solution Exercise 69 Evaluate the matrix product Solution Exercise 70 Evaluate the matrix product Solution Exercise 71 Evaluate the matrix product Solution All right, let's now turn to the inverse problems with unknown matrices.